A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

1, 2, 1, 2, 1, 3, 3, 4, 6, 3, 12, 10, 12, 14, 27, 38, 44, 52, 48, 77, 101, 106, 127, 206, 268, 377, 392, 496, 602, 671, 821, 1090, 1318, 1568, 1926, 2260, 2703, 3258, 3942, 4858, 5923, 6891, 8286, 9728, 11676, 13775, 16314, 19749, 23474, 27793, 32989, 38775

Offset: 1

Author

Gus Wiseman, Jan 15 2020

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
  1  2   3  4     5  6    7      8         9        A         B
     11     1111     222  3211   431       432      5311      542
                     321  22111  4211      3321     22111111  5411
                                 11111111  32211              33221
                                           321111             42221
                                           2211111            53111
                                                              322211
                                                              431111
                                                              521111
                                                              2222111
                                                              3311111
                                                              32111111
For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.
		

Crossrefs

The Heinz numbers of these partitions are given by A036844.

Numbers divisible by the sum of their prime indices are A324851.

Partitions whose product is divisible by their sum are A057568.

Partitions whose Heinz number is divisible by all parts are A330952.

Partitions whose Heinz number is divisible by their product are A324925.

Partitions whose Heinz number is divisible by their sum are A330950.

Partitions whose product is divisible by their sum of primes are A330954.

Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,Plus@@Prime/@#]&]],{n,30}]